Error bounds in local limit theorems using Stein's method
arXiv:1707.05995
Abstract
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős-Rényi random graph, and of the Curie-Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein's method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.
arXiv commentVer2: 28 pages, minor revision; Ver1: 27 pages
References in corpus (6)
- Applications of Stein's method for concentration inequalities
- Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing
- Translated Poisson approximation using exchangeable pair couplings
- Stein couplings for normal approximation
- Local limit theorems via Landau-Kolmogorov inequalities
- Discretized normal approximation by Stein's method